"Ara R. Aslyan" wrote:
>> On Fri, 14 Jan 2000, Mark Steiner wrote:
>> > As far as general intellectual interest of "core" mathematics, versus
> > philosophy, history, and foundations. Harvey asked me my view of their
> > view. There is no way really to know without an in-depth interview, but
> > my guess is (as I already said) that they would say that mathematics has
> > more intellectual interest than the history, philosophy, or foundations
> > of the field, if only because these other fields draw their intellectual
> > interest from core mathematics (of course, this is not an airtight
> > argument, as I pointed out above).
>> In order to express my disagreement with such a general point of
> view, let me consider the following example.
> Arithmetics is a study of various properties of natural numbers
> 0,0',0'', etc. while foudational studies (in Hilbert sense)
> roughly speaking concerned with the methods and principles being used in
> arithmetics, more precisely with the study of formal systems (PA,PRA,Z_o
> etc.). As it is well known much of arithmetics could be derived in this
> formal systems, using only a small part of proof theoretic strength of
> them. As this commented by P. Bernays any consistency proof for
> this formal systems would be a consistency proof "in advance" for a rather
> large possibilities than is needed to carry of an ordinary arithmetics.
> As it follows from Goedels results no such proof is possible within this
> systems (of course for the fixed formal system itself).
> Now the question is how the general intellectual interest of a
> subject to be "mesured". One way to do this (for me) to mesure the
> g.i.i. by the complxity of subject matter. With this respect, it is clear
> (at least for me) from the mentioned above that f.o.m. concerned with
> rather complicated issuers, and uses more complicated technics than
> is needed in ordinarry arithmetics.
> It would be interesting to hear your concept of g.i.i.. Ara Aslyan
I don't see any necessary connection between the complexity of subject
matter and the general intellectual interest that it has. But let's
analyze the case of Goedel's Theorem--namely the first one, since that
is regarded by all as having supreme g.i.i. Roger Penrose even wrote a
book about how you can argue from Goedel's first theorem to the
conclusion that human beings are not machines. Paul Benacerraf, in a
well known article that Penrose does not seem to have read, refuted this
argument long ago, but even Benacerraf thinks that something about the
human mind follows from Goedel's theorem (read the article). More to
the point something seems to follow about the concept of number from
Goedel's theorem, namely that it can't be exhausted by the usual
axiomatic deductive procedures of mathematics. Which is also about the
human mind. So the theorem has great intellectual interest, because of
its philosophical interest (a concept I personally feel more comfortable
with than g.i.i. of Harvey's). Goedel's first theorem also has a
PERCEIVED connnection with the Liar Paradox, a connection emphasized by
Goedel himself in the introduction to the proof. I think that paradoxes
have g.i.i., though I believe the Goedel misled many people by his
remarks about the Liar paradox, which probably were meant to be
heuristic.
Now the techniques used in proving Goedel's theorem itself are simply
those of elementary number theory, or primitive recursive functions. In
fact, that's what adds to its g.i.i., that it shows that the concepts of
number theory are much more complex, but it does so by using the same or
more elementary concepts.
Now, what is the g.i.i. of number theory? The Irish philosopher, George
Berkeley, stated (Principles of Human Knowledge, 119):
Arithmetic has been thought to have for its object abstract ideas of
Number; of
which to understand the properties and mutual habitudes, is supposed no
mean part of
speculative knowledge. The opinion of the pure and intellectual nature
of numbers in
abstract has made them in esteem with those philosophers who seem to
have affected an
uncommon fineness and elevation of thought. It hath set a price on the
most trifling
numerical speculations which in practice are of no use, but serve only
for amusement; and hath therefore so far infected the minds of some,
that they have dreamed of mighty
mysteries involved in numbers, and attempted the explication of natural
things by them.
But, if we inquire into our own thoughts, and consider what has been
premised, we may
perhaps entertain a low opinion of those high flights and abstractions,
and look on allinquiries, about numbers only as so many difficiles
nugae, so far as they are not subservient to practice, and promote the
benefit of life
I strongly disagree with this, but rather agree with Hardy, that beauty
has interest for its own sake. Mathematics has intellectual beauty,
particularly number theory, so mathematics has intellectual interest. I
think the beauty of mathematics (here I'm not speaking about number
theory in particular) and the peculiar connection of the beauty of
mathematics to its applicability gives it tremendous GENERAL
intellectual interest. I recall being amazed and inspired by Steven
Weinberg's remark that beautiful mathematics is eternal in physics,
meaning that even when theories in physics are refuted, their underlying
mathematical structures live on (Dirac's equation,for example). Now
beauty is not necessarily a matter of complexity; on the contrary, it's
simplicity which is often beautiful.
We see that foundations of mathematics and pure mathematics are
measured by different kinds of criteria.
I believe that both philosophy and f.o.m. can do a service to
themselves and to mathematics by make clear to the general public the
general intellectual interest of mathematics. They can probably do this
better than the mathematicians alone.